• thomas5267
Prove that the polynomials $$P_n$$ has n distinct root for all n. $$P_n$$ are characteristic polynomials of a particular type of matrix. $P_n=-xA_{n-1}-\frac{1}{2}A_{n-2}\\ A_n = \left( \dfrac{-x+\sqrt{x^2-1}}{2}\right)^n + \left(\dfrac{-x-\sqrt{x^2-1}}{2}\right)^n$ $$A_n$$ satisfies the following recurrence relation: $A_n=-x A_{n-1}-\frac{1}{4}A_{n-2}\\ A_1=-x\\ A_2=x^2-\frac{1}{2}$
Mathematics

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